Embedded newform 216.3.x.a.101.42

• Label: 216.3.x.a.101.42
• Level: $216$
• Weight: $3$
• Character: 216.101
• Analytic conductor: $5.886$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.88557371018\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.42
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.638953 + 1.89519i) q^{2} +(2.33253 - 1.88661i) q^{3} +(-3.18348 + 2.42187i) q^{4} +(-1.28960 - 7.31367i) q^{5} +(5.06586 + 3.21513i) q^{6} +(9.38727 + 7.87685i) q^{7} +(-6.62400 - 4.48583i) q^{8} +(1.88142 - 8.80115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.638953	+	1.89519i	0.319477	+	0.947594i
							\(3\)	2.33253	−	1.88661i	0.777511	−	0.628869i
\(5\)	−1.28960	−	7.31367i	−0.257919	−	1.46273i								−0.788466	−	0.615078i	\(-0.789124\pi\)
							0.530547	−	0.847656i	\(-0.321987\pi\)
\(7\)	9.38727	+	7.87685i	1.34104	+	1.12526i	0.981358	+	0.192189i	\(0.0615588\pi\)
							0.359681	+	0.933075i	\(0.382886\pi\)
\(11\)	−0.485568	+	2.75379i	−0.0441426	+	0.250345i	−0.998892	−	0.0470673i	\(-0.985012\pi\)
							0.954749	+	0.297412i	\(0.0961236\pi\)
\(13\)	3.34519	−	9.19084i	0.257322	−	0.706988i	−0.742008	−	0.670391i	\(-0.766126\pi\)
							0.999330	−	0.0365963i	\(-0.0116516\pi\)
\(17\)	23.4734	−	13.5524i	1.38079	−	0.797200i	0.388538	−	0.921433i	\(-0.372980\pi\)
							0.992253	+	0.124233i	\(0.0396469\pi\)
\(19\)	22.4249	+	12.9470i	1.18026	+	0.681423i	0.956074	−	0.293124i	\(-0.0946950\pi\)
							0.224184	+	0.974547i	\(0.428028\pi\)
\(23\)	−0.844509	−	1.00645i	−0.0367178	−	0.0437586i	0.747373	−	0.664405i	\(-0.231315\pi\)
							−0.784091	+	0.620646i	\(0.786870\pi\)
\(29\)	−45.6072	+	16.5997i	−1.57266	+	0.572402i	−0.973592	−	0.228295i	\(-0.926685\pi\)
							−0.599070	+	0.800697i	\(0.704463\pi\)
\(31\)	−31.8985	+	26.7660i	−1.02898	+	0.863421i	−0.990730	−	0.135847i	\(-0.956624\pi\)
							−0.0382550	+	0.999268i	\(0.512180\pi\)
\(37\)	−31.2770	+	18.0578i	−0.845325	+	0.488048i	−0.859071	−	0.511857i	\(-0.828958\pi\)
							0.0137459	+	0.999906i	\(0.495624\pi\)
\(41\)	3.87842	−	10.6559i	0.0945957	−	0.259899i	−0.883366	−	0.468684i	\(-0.844728\pi\)
							0.977962	+	0.208784i	\(0.0669506\pi\)
\(43\)	−24.0547	−	4.24149i	−0.559411	−	0.0986392i	−0.113208	−	0.993571i	\(-0.536113\pi\)
							−0.446202	+	0.894932i	\(0.647224\pi\)
\(47\)	−18.3586	+	21.8789i	−0.390608	+	0.465509i	−0.925132	−	0.379644i	\(-0.876046\pi\)
							0.534524	+	0.845153i	\(0.320491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.42	yes	420
8.5	even	2	inner	216.3.x.a.101.43	yes	420
27.23	odd	18	inner	216.3.x.a.77.43	yes	420
216.77	odd	18	inner	216.3.x.a.77.42	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.42	✓	420	216.77	odd	18	inner
216.3.x.a.77.43	yes	420	27.23	odd	18	inner
216.3.x.a.101.42	yes	420	1.1	even	1	trivial
216.3.x.a.101.43	yes	420	8.5	even	2	inner